Question

   Are the following functions from R to R? If not, explain why

a)    f(x) = 1/x

b)    f(x) = 2x + 7

Answer 1

Let us check whether the following correspondings are the functions from $\R$  to $\R.$

a)    $f(x) = \frac{1}{x}$

Since $f(x)$ is not defined for $x=0$, and for each $x\ne0$ the value $f(x)$ is uniquely determined, we conclude that $f$ is a function from $\R\setminus\{0\}$ to $\R,$ but $f$ is not a function from $\R$ to $\R.$

b)    $f(x) = 2x + 7$

Since for each $x\in \R$ the value $f(x)= 2x + 7$ is uniquely determined, we conclude that $f$ is a function from $\R$ to $\R.$